Middle Path table of eleven-limit rank two temperaments
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This table is supposed to be the best guess at what would have appeared in A Middle Path Part 2, had Paul Erlich finished it. It is similar to the Middle Path table of five-limit rank two temperaments and Middle Path table of seven-limit rank two temperaments from Part 1. It is intended to comprise all possible 11-limit 2D cases where complexity/7.65 + damage/10 < 1.
| Commas | Name | TOP
period |
TOP
generator |
Mapping | Cmplx | TOP
Dmg |
ETs |
| 36/35, 50/49, 64/63, 81/80, 126/125... | Duodecim | 99.59 | 31.48 | [<12 19 28 34 42|, <0 0 0 0 -1|] | 2.59 | 6.14 | 24d, 36d, 48cdde, (48cdd)... |
| 45/44, 56/55, 81/80, 100/99, 125/121... | Meanenneadecal | 1198.56 | 503.60 | [<1 2 4 7 6|, <0 -1 -4 -10 -6|] | 3.06 | 5.32 | 31e, 50ee, (62eee), 69dee, 81eee... |
| 36/35, 56/55, 64/63, 81/80, 99/98... | Dominant | 1195.04 | 495.58 | [<1 2 4 2 1|, <0 -1 -4 2 6|] | 3.08 | 4.96 | 29cde, 41cdee, 53cddeee... |
| 49/48, 56/55, 77/75, 121/120, 176/175... | Triforce | 399.80 | 155.57 | [<3 4 7 8 10|, <0 2 0 1 1|] | 3.22 | 5.29 | 39, 54cd, 69bcd, (78cd), 93bccdd... |
| 45/44, 50/49, 81/80, 99/98, 100/99... | Injera | 601.24 | 92.99 | [<2 3 4 5 6|, <0 1 4 4 6|] | 3.38 | 4.06 | 26, 38e, 50dee, (52c), 64ce, 66bcc... |
| 33/32, 49/48, 77/75, 99/98, 176/175... | Negric | 1205.30 | 129.26 | [<1 2 2 3 3|, <0 -4 3 -2 4|] | 3.52 | 5.30 | 28de, (56bddeee), (84bbdddeeeee)... |
| 56/55, 64/63, 100/99, 126/125, 176/175... | Augene | 398.88 | 87.10 | [<3 5 7 8 10|, <0 -1 0 2 2|] | 3.66 | 3.37 | 27e, 39dee, 42e, (54cee), 57bce... |
| 45/44, 49/48, 81/80, 100/99, 125/121... | Godzilla | 1204.09 | 254.92 | [<1 2 4 3 6|, <0 -2 -8 -1 -12|] | 3.71 | 4.09 | 33cd, 52cd, (66bccdd), 71bcdd... |
| 50/49, 55/54, 99/98, 100/99, 121/120... | Hedgehog | 599.09 | 163.15 | [<2 4 6 7 8|, <0 -3 -5 -5 -4|] | 3.75 | 3.23 | 22, 36ce, (44), 52bdd, 58ce, (66d)... |
| 49/48, 55/54, 77/75, 126/125, 176/175... | Darjeeling | 1202.79 | 318.16 | [<1 0 1 2 0|, <0 6 5 3 13|] | 3.81 | 4.41 | 34e, 53dee, (68dee), 83ddee... |
| 56/55, 64/63, 77/75, 121/120, 392/375... | Progress | 1195.64 | 559.36 | [<1 3 0 0 3|, <0 -3 5 6 1|] | 3.81 | 4.51 | 47bc, 62bcce, (94bbccce)... |
| 55/54, 64/63, 100/99, 121/120, 176/175... | Porcupine | 1198.23 | 163.15 | [<1 2 3 2 4|, <0 -3 -5 6 -4|] | 3.90 | 3.18 | 22, 37, (44), 52b, 59, (66d), 67b... |
| 49/48, 56/55, 100/99, 126/125, 343/330... | Keemun | 1200.82 | 318.18 | [<1 0 1 2 4|, <0 6 5 3 -2|] | 4.05 | 4.50 | 83dde, 117bddee, 151bdddeee... |
| 49/48, 81/80, 126/125, 225/224, 245/243... | Undevigintone | 63.36 | 30.42 | [<19 30 44 53 66|, <0 0 0 0 -1|] | 4.07 | 3.83 | 38d, 57dd, 57ddee, 76dd, (76dde)... |
| 50/49, 64/63, 99/98, 100/99, 176/175... | Pajara | 598.45 | 106.57 | [<2 3 5 6 8|, <0 1 -2 -2 -6|] | 4.17 | 3.11 | 22, 34d, (44), 46de, 56d, 58ddee... |
| 49/48, 55/54, 100/99, 121/120, 250/243... | Nautilus | 1202.66 | 82.97 | [<1 2 3 3 4|, <0 -6 -10 -3 -8|] | 4.21 | 3.48 | 29, (58cde), 72ccddee, 73cde... |
| 45/44, 81/80, 100/99, 125/121, 385/384... | Flattone | 1203.00 | 508.77 | [<1 2 4 -1 6|, <0 -1 -4 9 -6|] | 4.30 | 4.06 | 26, 45, (52c), 64cde, 71bc, (78bcc)... |
| 50/49, 55/54, 64/63, 225/224, 385/384... | Pajarous | 599.29 | 108.86 | [<2 3 5 6 6|, <0 1 -2 -2 5|] | 4.44 | 3.27 | 22, (44), (66d), (88bd), 98bcd... |
| 56/55, 81/80, 128/125, 176/175, 245/242... | Catnip | 99.81 | 31.73 | [<12 19 28 34 42|, <0 0 0 -1 -1|] | 4.48 | 3.56 | 36, 48ce, (72ce), 84cee, (96cceee)... |
| 55/54, 99/98, 176/175, 225/224, 245/243... | Telepathy | 1199.00 | 381.39 | [<1 0 2 -1 -1|, <0 5 1 12 14|] | 4.52 | 3.15 | 22, 41e, (44), 63e, (66d), (82eee)... |
| 55/54, 100/99, 121/120, 225/224, 250/243... | Porky | 1198.78 | 163.50 | [<1 2 3 5 4|, <0 -3 -5 -16 -4|] | 4.61 | 3.22 | 22, 29, (44), 51, (58cde), (66d)... |
| 56/55, 100/99, 126/125, 245/243, 540/539... | Sensis | 1196.58 | 442.49 | [<1 -1 -1 -2 2|, <0 7 9 13 4|] | 4.68 | 3.42 | 27e, (54cee), 73ee, (81bcdeee)... |
| 55/54, 64/63, 99/98, 245/243, 352/343... | Suprapyth | 1198.59 | 490.29 | [<1 2 6 2 1|, <0 -1 -9 2 6|] | 4.68 | 3.18 | 22, (44), (66d), (88bd), 93bde... |
| 50/49, 64/63, 225/224, 245/243, 250/243... | Vigintiduo | 54.48 | 10.51 | [<22 35 51 62 76|, <0 0 0 0 1|] | 4.74 | 3.28 | 66de, 88bde, 110bdd, 110bddee... |
| 81/80, 99/98, 126/125, 176/175, 225/224... | Meantone | 1201.61 | 504.02 | [<1 2 4 7 11|, <0 -1 -4 -10 -18|] | 4.96 | 1.74 | 31, 43, 50e, 55de, (62), 74, 81ee... |
| 50/49, 55/54, 176/175, 352/343, 540/539... | Fleetwood | 599.28 | 272.31 | [<2 5 6 7 11|, <0 -4 -3 -3 -9|] | 4.96 | 3.27 | 22, (44), (66d), (88bd), (110bdde)... |
| 50/49, 121/120, 176/175, 352/343, 385/384... | Astrology | 599.32 | 217.89 | [<2 5 5 6 8|, <0 -5 -1 -1 -3|] | 5.07 | 3.28 | 22, (44), (66d), (88bd), (110bdde)... |
| 49/48, 55/54, 225/224, 441/440, 525/512... | Negroni | 1203.19 | 124.84 | [<1 2 2 3 5|, <0 -4 3 -2 -15|] | 5.11 | 3.19 | 77cddee, 106ccdddeee... |
| 64/63, 99/98, 121/120, 352/343, 540/539... | Quasisupra | 1197.34 | 490.80 | [<1 2 -3 2 1|, <0 -1 13 2 6|] | 5.29 | 2.66 | 39d, 61d, (78cdd), 83d, 100bcdd... |
| 99/98, 121/120, 176/175, 225/224, 385/384... | Orwell | 1201.25 | 271.43 | [<1 0 3 1 3|, <0 7 -3 8 2|] | 5.30 | 1.36 | 31, 53, (62), 75, 84e, (93e), 97de... |
| 81/80, 99/98, 121/120, 441/440, 540/539... | Squares | 1201.70 | 426.46 | [<1 3 8 6 7|, <0 -4 -16 -9 -10|] | 5.37 | 1.70 | 31, 45e, (62), 76e, 79c, 90bcdee... |
| 64/63, 100/99, 176/175, 245/243, 540/539... | Superpyth | 1197.60 | 489.43 | [<1 2 6 2 10|, <0 -1 -9 2 -16|] | 5.42 | 2.40 | 49, 71d, 76bcdee, 93bd, (98bde)... |
| 121/120, 126/125, 176/175, 385/384, 441/440... | Valentine | 1200.87 | 77.63 | [<1 1 2 3 3|, <0 9 5 -3 7|] | 5.82 | 1.54 | 31, 46, (62), 77, (92), (93e), 107c... |
| 81/80, 121/120, 176/175, 243/242, 385/384... | Mohajira | 1201.70 | 348.78 | [<1 1 0 6 2|, <0 2 8 -11 5|] | 6.01 | 1.70 | 31, (62), (93e), (124be), 148be... |
| 100/99, 225/224, 245/243, 385/384, 540/539... | Magic | 1200.75 | 380.92 | [<1 0 2 -1 6|, <0 5 1 12 -8|] | 6.07 | 1.68 | 41, 63, (82), 104, (123c), (126c)... |
| 81/80, 126/125, 225/224, 385/384, 540/539... | Meanpop | 1201.70 | 504.13 | [<1 2 4 7 -2|, <0 -1 -4 -10 13|] | 6.08 | 1.70 | 31, 50, (62), 81, (93e), (100d), 112b... |
| 81/80, 121/120, 126/125, 225/224, 243/242... | Migration | 1201.70 | 348.78 | [<1 1 0 -3 2|, <0 2 8 20 5|] | 6.10 | 1.70 | 31, (62), (93e), 100de, (124be)... |
| 99/98, 121/120, 126/125, 540/539, 891/875... | Nusecond | 1200.46 | 154.74 | [<1 3 4 5 5|, <0 -11 -13 -17 -12|] | 6.13 | 1.70 | 31, (62), (93e), 101, (124be), 132ce... |
| 100/99, 225/224, 245/242, 441/440, 625/616... | Andromeda | 1201.36 | 497.94 | [<1 2 -1 -3 -4|, <0 -1 8 14 18|] | 6.19 | 1.79 | 152cd, 193ccd, 234ccd, 263cccdd... |
| 225/224, 243/242, 385/384, 441/440, 540/539... | Miracle | 1200.63 | 116.72 | [<1 1 3 3 2|, <0 6 -7 -2 15|] | 7.14 | 0.63 | 72, (144), (216c), (288cd), (360bcd)... |
| A BONUS TEMPERAMENT: | |||||||
| 2401/2400, 3025/3024, 4375/4374, 9801/9800... | Hemiennealimmal | 66.67 | 17.64 | [<18 28 41 50 62|, <0 2 3 2 1|] | 23.81 | 0.05 | |
| 36/35, 50/49, 64/63, 81/80, 126/125... | Duodecim | 99.59 | 31.48 | [<12 19 28 34 42|, <0 0 0 0 -1|] | 2.59 | 6.14 | 24d, 36d, 48cdde, (48cdd)... |
| 56/55, 81/80, 128/125, 176/175, 245/242... | Catnip | 99.81 | 31.73 | [<12 19 28 34 42|, <0 0 0 -1 -1|] | 4.48 | 3.56 | 36, 48ce, (72ce), 84cee, (96cceee)... |
Since the TOP tunings of duodecim and catnip are so similar, there is little practical use for duodecim temperament. If notes so altered from 12edo are available, there is no reason not to use them for ratios of 7 as well as ratios of 11.
| 45/44, 56/55, 81/80, 100/99, 125/121... | Meanenneadecal | 1198.56 | 503.60 | [<1 2 4 7 6|, <0 -1 -4 -10 -6|] | 3.06 | 5.32 | 31e, 50ee, (62eee), 69dee, 81eee... |
| 36/35, 56/55, 64/63, 81/80, 99/98... | Dominant | 1195.04 | 495.58 | [<1 2 4 2 1|, <0 -1 -4 2 6|] | 3.08 | 4.96 | 29cde, 41cdee, 53cddeee... |
| 45/44, 81/80, 100/99, 125/121, 385/384... | Flattone | 1203.00 | 508.77 | [<1 2 4 -1 6|, <0 -1 -4 9 -6|] | 4.30 | 4.06 | 26, 45, (52c), 64cde, 71bc, (78bcc)... |
| 81/80, 99/98, 126/125, 176/175, 225/224... | Meantone | 1201.61 | 504.02 | [<1 2 4 7 11|, <0 -1 -4 -10 -18|] | 4.96 | 1.74 | 31, 43, 50e, 55de, (62), 74, 81ee... |
| 81/80, 126/125, 225/224, 385/384, 540/539... | Meanpop | 1201.70 | 504.13 | [<1 2 4 7 -2|, <0 -1 -4 -10 13|] | 6.08 | 1.70 | 31, 50, (62), 81, (93e), (100d), 112b... |
These temperaments all temper out 81/80, making them extensions of 5-limit meantone. The differences are in the mappings of 7 and 11.
| 36/35, 56/55, 64/63, 81/80, 99/98... | Dominant | 1195.04 | 495.58 | [<1 2 4 2 1|, <0 -1 -4 2 6|] | 3.08 | 4.96 | 29cde, 41cdee, 53cddeee... |
| 55/54, 64/63, 99/98, 245/243, 352/343... | Suprapyth | 1198.59 | 490.29 | [<1 2 6 2 1|, <0 -1 -9 2 6|] | 4.68 | 3.18 | 22, (44), (66d), (88bd), 93bde... |
| 64/63, 99/98, 121/120, 352/343, 540/539... | Quasisupra | 1197.34 | 490.80 | [<1 2 -3 2 1|, <0 -1 13 2 6|] | 5.29 | 2.66 | 39d, 61d, (78cdd), 83d, 100bcdd... |
| 64/63, 100/99, 176/175, 245/243, 540/539... | Superpyth | 1197.60 | 489.43 | [<1 2 6 2 10|, <0 -1 -9 2 -16|] | 5.42 | 2.40 | 49, 71d, 76bcdee, 93bd, (98bde)... |
In contrast, these temperaments all temper out 64/63, making them extensions of archy temperament. The differences are in the mappings of 5 and 11. Temperaments in both this and the above list comprise the dominant family, which has only one representative here. Note that cassandra is in neither of the two families, because it preserves both 64/63 and 81/80 as non-vanishing intervals (and makes them both equal to the Pythagorean comma).
| 33/32, 49/48, 77/75, 99/98, 176/175... | Negric | 1205.30 | 129.26 | [<1 2 2 3 3|, <0 -4 3 -2 4|] | 3.52 | 5.30 | 28de, (56bddeee), (84bbdddeeeee)... |
| 49/48, 55/54, 225/224, 441/440, 525/512... | Negroni | 1203.19 | 124.84 | [<1 2 2 3 5|, <0 -4 3 -2 -15|] | 5.11 | 3.19 | 77cddee, 106ccdddeee... |
The difference is only in the mapping of 11. The two temperaments intersect in 19edo (using the 19e val tempering out 33/32), which is a fine tuning for negric (despite that it doesn't show up in the list), but sub-optimal for negroni (which does not temper out 33/32).
| 50/49, 64/63, 99/98, 100/99, 176/175... | Pajara | 598.45 | 106.57 | [<2 3 5 6 8|, <0 1 -2 -2 -6|] | 4.17 | 3.11 | 22, 34d, (44), 46de, 56d, 58ddee... |
| 50/49, 55/54, 64/63, 225/224, 385/384... | Pajarous | 599.29 | 108.86 | [<2 3 5 6 6|, <0 1 -2 -2 5|] | 4.44 | 3.27 | 22, (44), (66d), (88bd), 98bcd... |
The difference is only in the mapping of 11. The two temperaments intersect in 22edo, a fine tuning for both.
| 49/48, 55/54, 77/75, 126/125, 176/175... | Darjeeling | 1202.79 | 318.16 | [<1 0 1 2 0|, <0 6 5 3 13|] | 3.81 | 4.41 | 34e, 53dee, (68dee), 83ddee... |
| 49/48, 56/55, 100/99, 126/125, 343/330... | Keemun | 1200.82 | 318.18 | [<1 0 1 2 4|, <0 6 5 3 -2|] | 4.05 | 4.50 | 83dde, 117bddee, 151bdddeee... |
The difference is only in the mapping of 11. The two intersect in 15edo, which is, however, not a great tuning for either. In 19edo, darjeeling uses the 19e val, which tempers out 33/32, whereas keemun uses the patent val.
| 55/54, 64/63, 100/99, 121/120, 176/175... | Porcupine | 1198.23 | 163.15 | [<1 2 3 2 4|, <0 -3 -5 6 -4|] | 3.90 | 3.18 | 22, 37, (44), 52b, 59, (66d), 67b... |
| 55/54, 100/99, 121/120, 225/224, 250/243... | Porky | 1198.78 | 163.50 | [<1 2 3 5 4|, <0 -3 -5 -16 -4|] | 4.61 | 3.22 | 22, 29, (44), 51, (58cde), (66d)... |
The difference is only in the mapping of 7. The two intersect in 22edo, which is probably the only reasonable incarnation of porky temperament.
| 55/54, 99/98, 176/175, 225/224, 245/243... | Telepathy | 1199.00 | 381.39 | [<1 0 2 -1 -1|, <0 5 1 12 14|] | 4.52 | 3.15 | 22, 41e, (44), 63e, (66d), (82eee)... |
| 100/99, 225/224, 245/243, 385/384, 540/539... | Magic | 1200.75 | 380.92 | [<1 0 2 -1 6|, <0 5 1 12 -8|] | 6.07 | 1.68 | 41, 63, (82), 104, (123c), (126c)... |
The difference is only in the mapping of 11. The two intersect in 22edo, which is a fine telepathy tuning but slightly sub-optimal for magic. Since telepathy is significantly higher in error, it can be regarded as an alternate version of magic that exists in 22edo.
| 81/80, 121/120, 176/175, 243/242, 385/384... | Mohajira | 1201.70 | 348.78 | [<1 1 0 6 2|, <0 2 8 -11 5|] | 6.01 | 1.70 | 31, (62), (93e), (124be), 148be... |
| 81/80, 121/120, 126/125, 225/224, 243/242... | Migration | 1201.70 | 348.78 | [<1 1 0 -3 2|, <0 2 8 20 5|] | 6.10 | 1.70 | 31, (62), (93e), 100de, (124be)... |
The TOP tunings of mohajira and migration are not merely close, but exactly equal, because the prime 7 does not affect the TOP tuning. The two temperaments intersect in 31edo, which is also near-optimal for both.
| ET | TOP damage | TOP octave | Rank-2 temperaments supported |
| 22 | hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology | ||
| 24d | duodecim | ||
| 26 | injera, flattone | ||
| 27e | augene, sensis | ||
| 28de | negric | ||
| 29 | nautilus, porky | ||
| 29cde | dominant | ||
| 31 | meantone, orwell, squares, valentine, mohajira, meanpop, migration, nusecond | ||
| 31e | meanenneadecal | ||
| 33cd | godzilla | ||
| 34d | pajara | ||
| 34e | darjeeling | ||
| 36 | catnip | ||
| 36ce | hedgehog | ||
| 36d | duodecim | ||
| 37 | porcupine | ||
| 38d | undevigintone | ||
| 38e | injera | ||
| 39 | triforce | ||
| 39d | quasisupra | ||
| 39dee | augene | ||
| 41 | magic | ||
| 41cdee | dominant | ||
| 41e | telepathy | ||
| 42e | augene | ||
| 43 | meantone | ||
| 44* | hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology | ||
| 45 | flattone | ||
| 45e | squares | ||
| 46 | valentine | ||
| 46de | pajara | ||
| 47bc | progress | ||
| 48cdd* | duodecim | ||
| 48cdde | duodecim | ||
| 48ce | catnip | ||
| 49 | superpyth | ||
| 50 | meanpop | ||
| 50dee | injera | ||
| 50e | meantone | ||
| 50ee | meanenneadecal | ||
| 51 | porky | ||
| 52b | porcupine | ||
| 52bdd | hedgehog | ||
| 52c* | injera, flattone | ||
| 52cd | godzilla | ||
| 53 | orwell | ||
| 53cddeee | dominant | ||
| 53dee | darjeeling | ||
| 54cd | triforce | ||
| 54cee* | augene, sensis | ||
| 55de | meantone | ||
| 56bddeee* | negric | ||
| 56d | pajara | ||
| 57bce | augene | ||
| 57dd | undevigintone | ||
| 57ddee | undevigintone | ||
| 58cde* | nautilus, porky | ||
| 58ce | hedgehog | ||
| 58ccddee* | dominant | ||
| 58ddee | pajara | ||
| 59 | porcupine | ||
| 60cddd | duodecim | ||
| 60cdddee | duodecim | ||
| 61d | quasisupra | ||
| 62* | meantone, orwell, squares, valentine, mohajira, meanpop, migration, nusecond | ||
| 62bcce | progress | ||
| 62eee* | meanenneadecal | ||
| 63 | magic | ||
| 63e | telepathy | ||
| 64cd | injera | ||
| 64cde | flattone | ||
| 65ccdddeeee | dominant | ||
| 66bcc | injera | ||
| 66bccdd* | godzilla | ||
| 66d* | hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology | ||
| 66de | vigintiduo | ||
| 67b | porcupine | ||
| 68dde* | pajara | ||
| 68dee* | darjeeling | ||
| 69bcd | triforce | ||
| 69bcee | augene | ||
| 69dee | meanenneadecal | ||
| 70ccddeee | dominant | ||
| 70ddeee | pajara | ||
| 71bc | flattone | ||
| 71bcdd | godzilla | ||
| 71d | superpyth | ||
| 72 | miracle | ||
| 72bce | augene | ||
| 72ccddee | nautilus | ||
| 72ccee* | hedgehog | ||
| 72cddd* | duodecim | ||
| 72cddde* | duodecim | ||
| 72cdddeee | duodecim | ||
| 72ce* | catnip |